Absolutely Not

As others have stated, no you can not use a focus (of any kind) while Wild Shaped, even if the animal you morph into has hands, and they are empty and holding the focus. All of the information you posted in the original question provides the answers for you so long as you cross reference them. But a lot of people have missed a crucial part of why it can't be done.

The description of the Druidic Focus says (5e PHB, p. 151):

A druidic focus might be [...] a totem object incorporating feathers, fur, bones, and teeth from sacred animals. A druid can use such an object as a spellcasting focus.

This essentially states that you get to create a focus out of whatever you deem worthy to call a focus; pretty neat.

The description of Material (M) components says:

Casting some spells requires particular objects, specified in parentheses in the component entry. A character can use [...] a spellcasting focus in place of the components specified for a spell [but not costly or consumed components]

This is the start of the unravel. Here it says you can use a focus instead of a material component (so long as it has no cost/isn't consumed). So essentially focuses are just fluff/filler for material components that most people over look anyway, but if you don't overlook that stuff it's still a pretty minimal net gain.

Part of the description of the druid's Wild Shape feature says (5e PHB, p. 67):

• You choose whether your equipment falls to the ground in your space, merges into your new form, or is worn by it. [...]

This portion infers that you get to wear or hold your focus if you so choose when you Wild Shape, which is pretty cool.

The druid's Beast Spells feature says (5e PHB, p. 67):

Beginning at 18th level, you can cast many of your druid spells in any shape you assume using Wild Shape. You can perform the somatic and verbal components of a druid spell while in a beast shape, but you aren't able to provide material components.

This is the crucial spot. It specifically says no material components. If you recall from what exactly a focus is, it is essentially a fancy material component. Meaning it can't be used, not because you can't "hold" it, but because for Game Design purposes they disallowed it.

This is further backed up by the fact that they allow somatic components (hand and body gestures) even if you don't have hands or the appropriate body part to do the somatic component, i.e. a snake making a hand sign. So holding/wearing the material component/focus is irrelevant because it is completely outlawed in RAW.

The druid's Archdruid capstone feature says (emphasis mine):

At 20th level, you can use your Wild Shape an unlimited number of times.

Additionally, you can ignore the verbal and somatic components of your druid spells, as well as any material components that lack a cost and aren't consumed by a spell. You gain this benefit in both your natural shape and your beast shape from Wild Shape.

Doesn't this seem a bit redundant? It is because it is trying to show you that it is essentially an improvement to Beast Spells. At this point, once you reach 20th level you no longer need the material/focus component. So even at level 20 the answer is still technically no, but it becomes irrelevant because materials/focuses are no longer even needed.

However, this is D&D - so if the DM says, "Well that's stupid, you should be able to use your focus as long as you are wearing/holding it"... Then boom, there you go: the true answer is "yes, you can, so long as your DM handwaves it".

Innate spells only ignore the components that are explicitly listed

Innate spells follow the normal rules for spell components unless otherwise noted

An innate spell can have special rules or restrictions. For example, a drow mage can innately cast the levitate spell, but the spell has a “self only” restriction, which means that the spell affects only the drow mage. (MM 10)

Since innate spells are simply normal spells¹, they follow normal spellcasting rules unless there is a specific rule that overrides the general. This means that innate spells need all the components as normal spells unless a specific rule for that monster says otherwise.

Jeremy Crawford has confirmed this via Twitter (several times):

Q: Does innate spell casting use V and S components?

A: A creature's Innate Spellcasting trait tells you if it doesn't require certain components. It otherwise requires components.

Only innate spellcasting that ignores all components that are required by the spell is not able to be counterspelled

This does not conflict at all with the passage you quote:

If the need for a spell’s components has been removed by a special ability, such as the sorcerer’s Subtle Spell feature or the Innate Spellcasting trait possessed by many creatures, the casting of the spell is imperceptible.

All this is saying that the trait, innate spellcasting, is possessed by many creatures and is an example of a trait that can potentially make a spell imperceptible. It never implies innate spellcasting automatically makes a spell imperceptible by removing the need for any components. As shown above, that is not at all how the rules say it works.

This is even more easily seen by the fact that it lists subtle spell as well. Since subtle spell explicitly only removes any somatic or verbal components it certainly is not an example of an ability that automatically makes a spell imperceptible. If a spell were to have material components it would still be perceptible.

Thus, only if an innate spell specifically says that no components need to be used for the spell, is that spell unable to be counterspelled.

Examples

Many (if not all) innate spellcasting traits specify that the spells do not require material components. This is specified in the monster statblocks:

The mephit can innately cast dancing lights, requiring no material components. Its innate spellcasting ability is Charisma.

So this spell needs no material components since the ability specifically overrides the general rule. However, since the ability says nothing about verbal or somatic components (which dancing lights requires), those are still required as per the general spellcasting rules.

However the Githyanki Warrior's ability is different:

The githyanki’s innate spellcasting ability is Intelligence. It can innately cast the following spells, requiring no components...

Since it says no components are required, these spells would be imperceptible and thus not counterspellable.

^{1 - The Monster Manual describes innate spellcasting as a monster ability with which they can cast spells:}

^{A monster with the innate ability to cast spells has the Innate Spellcasting special trait.}

^{Jeremy Crawford also explicitly supports this:}

^{A spell is a spell no matter how you cast it: spellcasting, innate spellcasting, racial trait, magic item, class feature, etc.}